G.R.Dixon INTRODUCTION. This monograph discusses the quantum theory of a finite square well. It employs the spinning spiral model discussed in a previous article. A brief review of that model is provided in Sect. 1. The spinning spiral model is analogous to a circularly polarized electromagnetic wave when such a wave propagates in the 1. REVIEW. Zero Potential Energy. When an ensemble of monoenergetic particles moves in the positive or negative x-direction, a rotating spiral provides a convenient model for visualizing the quantum wave function in space and time. A left-handed spiral, rotating clockwise (looking in the positive x-direction), can represent the wave function for particles moving in the positive x-direction. Similarly, a right-handed spiral (also rotating clockwise) can represent the wave function for particles moving in the negative x-direction. The threads of the left-handed spiral move in the positive x-direction as the spiral spins; the threads of the right-handed spiral move in the negative x-direction. At any given x the wave function for either particle ensemble can be visualized as a vector in a complex plane oriented orthogonal to the x-axis and with its origin on the x-axis. The vector’s tail would then coincide with the x-axis, and its head would be attached to the associated spiral. As previously mentioned, with no loss of rigor the complex plane’s real axis can be aligned parallel to the z-axis, and its imaginary axis can be aligned parallel to the y-axis. The formulas for the wave function of particles traveling toward positive (Y , (1.1a) . (1.1b) Note that at times t other than zero, Y , (1.1c) . (1.1d) The real constant k (the wave number) is related to either spiral’s wavelength (l) by: . (1.2a) And the real constant w is related to either spiral’s spin frequency (n) by: . (1.2b) DeBroglie suggested that the wavelength is related to particle momentum magnitude by . (1.3a) He also suggested that the spiral’s spin frequency is related to the energy (which equals the particle kinetic energy when the potential energy, U, is zero): . (1.3b) ln, the spiral thread speed, is then half the particle speed: . (1.4) Born suggested that the squared magnitude of the wave function (which is always a real number) is the probability density for finding a particle at a given location. For example, the probability of finding a particle traveling in the positive x-direction and in the interval x to x+dx (or (x,dx) for short) is: . (1.5) Note that P in this simple example is actually independent of x (and t). That is, the
magnitude of Y Normalization … the requirement that the integral of Pdx over all x be equal to unity (i.e., the requirement that the particle will be found
somwhere) … indicates that |Y The first "quantum surprise" occurs when two such ensembles are combined. For in this case the probability of finding a particle traveling in either direction, in some interval (x,dx), is . (1.6) With no loss of rigor we can let
Y ^{
.
(1.7)
}
It is noteworthy that
^{
.
(1.8)
}This consequence of adding Y Infinite Barriers. Fig. 1.1 depicts an infinite potential energy barrier at x=0. U is zero for all x<0, and U is infinite for all x>0. Both classically and in quantum theory, particles traveling along the negative x-axis and in the positive x-direction are turned back at x=0. Figure 1.1
An Infinite Potential Energy Barrier Since all particles are turned back at x=0, P(x>0)=0. And of course P(x<0) is proportional to sine squared and is given by Eq. 1.8. It is logically plausible that P must be continuous in x. This being the case, we can again specify with no loss of rigor that , (1.9a) . (1.9b) ^{
}
Square Wells. Fig. 1.2 depicts an infinite square well. Each wall constitutes an infinite barrier to particles trapped in the well. Particles traveling in the positive x-direction are turned back at x=L; those traveling in the negative x-direction are turned back at x=0. Assuming the particles are monoenergetic, Eqs. 1.1c and d again specify the wave functions for the two ensembles. And of course Eq. 1.7 gives their sum. Figure 1.2
An Infinite Square Potential Energy Well The requirement that P be continuous in x can be satisfied at x=0 by again stipulating that
Y ^{
(1.10)
}According to deBroglie, this means that only discrete momentum magnitudes (and thus only discrete kinetic energies) can be accommodated by an infinite well. For example, the lowest energy (or "ground") state must have a wavelength of l/2: ^{
.
(1.11)
But according to deBroglie,
.
(1.12)
}Thus the ground state energy, E . (1.13) Similarly, , (1.14) In general: . (1.15) Let us trace a complete circuit from x=0 to x=L and back again. We shall consider the ground state, where L equals a half wavelength. By the time we have moved along the left-handed spiral for
Y Any wavelength not satisfying
Eq. 1.10 will not produce unique values of Y ^{
2. A STRATEGY FOR SOLVING THE FINITE WELL.
}
Fig. 2.1 depicts a
finite potential energy well. We shall attempt to solve for Y Figure 2.1 ^{
}
A Finite Square Well For particles trapped in the well and traveling in the negative x-direction, the wall at x=0 constitutes a finite barrier. Similarly for particles traveling in the positive x-direction when they encounter the wall at x=L. It thus behooves us to begin our analysis of the finite well by considering the wave function for particles that encounter a finite barrier. 3. THE FINITE BARRIER. Fig. 3.1 depicts a finite potential barrier at x=0. Monoenergetic particles, with kinetic energy mv Figure 3.1 ^{
}
A Finite Potential Energy Barrier We can satisfy the requirement that |Y(0)| be nonzero by stipulating that
Y Figure 3.2 ^{
}
and Y_{(-)}(0)
^{
}
In Fig. 3.2 we have conveniently stipulated that ^{
,
(3.1a)
.
(3.1b)
Thus for x>0,
,
(3.2a)
.
(3.2b)
Summing Eqs. 3.2a and b, we find that
(3.3)
.
And
,
(3.4)
.
(3.5)
}Or, since sin(2q)=2 sin(q)cos(q), . (3.6) Specifically, at x=0: , (3.7a) . (3.7b) In considering the situation at negative x, we might begin by assuming that E, the total energy, is a system constant: . (3.8) Since U(x>0)=0, the total energy equates to mv , (3.9a) . (3.9b) By deBroglie, the wave numbers for Y’ , (3.10a) . (3.10b) Substituting in Eqs. 3.2a and b produces: , (3.11a) . (3.11b) Or, defining a to be: , (3.12) Eqs. 3.11a and be can be written as: , (3.13a) . (3.13b) Note that neither Y’ Summing Eqs. 3.13a and b produces: . (3.14) Therefore for x<0: . (3.15a) Differentiation produces: . (3.15b) At x=0: , (3.16a) . (3.16b) Comparison of Eqs. 3.7a and 3.16a shows that P is indeed continuous at x=0. And (see Eqs. 3.7b and 3.16b), the continuity of dP/dx at x=0 requires that . (3.17) Or, . (3.18) Now by definition, , (3.19) and thus . (3.20) Therefore . (3.21) Eq. 3.18 can therefore be rewritten as: . (3.22) Finally, . (3.23) As expected, in the limit of infinite U Like an infinite barrier, a finite barrier will reflect all particles with kinetic energies less than the barrier height. Or, by
deBroglie, all wave functions of particles meeting this condition will be reflected. However, in the finite barrier case the wave function penetrates into the forbidden zone. It is partially reflected in each interval (x<0,dx). Consequently the phase difference between Y’ 4. THE FINITE SQUARE WELL. Let us now consider the finite square well (see Fig. 2.1). It will be instructive to consider the first few energy states individually. We expect something less than half a wavelength to be contained in the Ground State
(E=E The Ground State Fig. 4.1 qualitatively depicts values for
Y Figure 4.1 ^{
}
Y ^{
}
The requirement that
Y In a traversal of Y . (4.1) Or, . (4.2) But . (4.3) Thus . (4.4) This transcendental equation in E The First Excited State. Fig. 4.2 qualitatively depicts suggested values for Y Figure 4.2
First Excited State The angle f
depends only on U , (4.5) or . (4.6) In this case , (4.7) whence . (4.8) Again we can "creep backward" from the infinite well energy of h Second Excited State. Fig. 4.1 also depicts suggested values of Y , (4.9) and . (4.10) Thus . (4.11) Arbitrary E For arbitrary E . (4.12) 5. Some Concluding Thoughts.
The representation of Y
(in one dimension) as the sum of
Y It is interesting that |Y| In real world temperatures above absolute zero a large set of systems will, at any particular moment, usually have some systems at energy E All told there are many lines of inquiry where the spinning spiral model might lead to better understanding and even to new insights. The complex algebra/calculus of wave mechanics may at times seem daunting. Yet practically anyone can visualize a rotating spiral in his/her mind’s eye. Comments? mailto:noxid100@cox.net ^{
} |